## An Introduction to Default LogicThis book is written for those who are interested in a fonnalization of human reasoning, especially in order to build "intelligent" computer systems. Thus, it is mainly designed for the Artificial Intelligence community, both students and researchers, although it can be useful for people working in related fields like cognitive psychology. The major theme is not Artificial Intelligence applications, although these are discussed throughout in sketch fonn. Rather, the book places a heavy emphasis on the fonnal development of default logic, results and problems. Default logic provides a fonnalism for an important part of human reasoning. Default logic is specifically concerned with common sense reasoning, which has recently been recognized in the Artificial Intelligence literature to be of fundamental importance for knowledge representation. Previously, fonnalized reasoning systems failed in real world environments, though succeeding with an acceptable ratio in well-defined environments. This situation enabled empirical explorations and the design of systems without theoretical justification. In particular, they could not be compared since there was no basis to judge their respective merits. Default logic turned out to be very fruitful by proving the correctness of some of them. We hope that this book will initiate other successful developments in default logic. |

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### Contents

Default Reasoning | 1 |

12 Default Reasoning | 3 |

Formal Logic as a Knowledge Representation Scheme | 5 |

Syntactical Part | 6 |

Semantical Part | 8 |

24 Properties for Formal Logic | 9 |

25 Logical Languages for Knowledge Representation | 11 |

First Order Logic | 13 |

Default Logic Revisited | 111 |

112 Modified Extensions | 113 |

113 Lukaszewicz ProofTheoretic Approach to Default Logic | 119 |

114 Lukaszewicz ModelTheoretic Approach to Default Logic | 123 |

115 The BeliefsJustifications Pairs Setting | 124 |

116 Default Logic Revisited Versus Default Logic? | 128 |

Circumscription | 131 |

122 Model Theory for Predicate Circumscription | 135 |

32 First Order Model Theory | 19 |

33 First Order Proof Theory | 22 |

34 Properties of First Order Logic | 25 |

Nonmonotonic Extensions for First Order Logic | 27 |

42 Retaining the Advantages of First Order Logic | 29 |

Presentation of Default Logic | 31 |

Formal Development of Default Logic | 37 |

62 General Properties of Default Theories | 44 |

Normal Defaults | 53 |

72 Default Proof Theory | 60 |

73 Default Model Theory | 67 |

74 Completeness and Decidability Matters | 74 |

Further Topics in Default Logic | 75 |

82 Interacting Defaults | 80 |

Fragments of Default Logic | 89 |

92 SemiNormal Default Theories | 94 |

93 Taxonomic Default Theories | 99 |

Problems with Default Logic | 101 |

102 Undesirable Features of Default Logic | 108 |

123 Existence of Minimal Models | 141 |

124 Equality and Predicate Circumscription | 146 |

125 Joint Predicate Circumscription | 149 |

126 First Order Formula Circumscription | 153 |

127 Second Order Formula Circumscription | 155 |

128 Prioritized Formula Circumscription | 158 |

129 Pointwise Circumscription | 160 |

Other Logic Formalizations of Nonmonotonic Reasoning | 163 |

132 Intuitionistic Nonmonotonic Logic | 167 |

133 Modal Nonmonotonic Logic for Axiomatic Default Theories | 170 |

134 ThreeValued Nonmonotonic Logic | 172 |

135 Autoepistemic Logic | 174 |

136 The Logic of Theory Change | 189 |

137 Logic Systems for Belief Revision | 192 |

Origin of the Theorems | 193 |

195 | |

Table of Symbols | 203 |

205 | |

### Common terms and phrases

Artificial Intelligence autoepistemic logic axiom schema axiomatic theory believe Besnard BJ-pair Chapter circumscription schema Closed World Assumption Consider consistent contradiction corresponding CT[P CUBE deduction theorem deductively closed default logic default proof default reasoning definable with parameters Definition ELEPHANT Etherington exists finite FLY Tweety formal consequence Formal Logic free variables given Herbrand inconsistent inference rules instance intuitionistic knowledge representation lemma Lukaszewicz means minimal model modal logic S5 modal operator model 9 model theory modified extension modus ponens NML1 node nonmonotonic logic nonmonotonic reasoning normal default theory notion order formula circumscription order interpretation order language order logic predicate circumscription predicate symbol proof theory Proposition 6.2.6 propositional logic Reiter Reiter Reiter respect result satisfies schemata semantical semi-normal default theories sequence set of defaults set of formulas set of justifications set of sentences Skolem stable expansion subset syntactical Th(T Theorem truth value UNHAPPY John unique extension universal theory